Folks - Please try this, based on a well-worn puzzle. Please read the final part before firing back - thanks - MmM
25 gnomes have been captured by an evil king. The King tells them he will line up the gnomes in a single-file row, so that each can see the backs of those in front of him; thus the rearmost gnome will see 24 backs and the gnome in front will see no backs.
The king continues that he will randomly distribute black and white hats, placing one on each of the gnomes’ heads so that none can see his own hat, nor those behind him, but each can see all hats before him.
The callous king will move along the line from the rear, asking each gnome ‘What colour is your hat?’ Any gnome who answers incorrectly will be shot immediately. So each gnome needs to determine, as accurately as possible, the colour of his own hat, based on those he sees before him and the declarations / shots he hears from behind him.
The King allows them to determine a strategy beforehand, with the aim of saving as many gnomic lives as possible. Assume that all gnomes are alike in terms of being logically-minded, with sufficient eyesight and hearing to cover all 25 gnomes. A gnome may only state his calculated hat colour in a level tone of voice. He may not ‘pass’. If they say or do anything else, the king will shoot all gnomes. This is a well-known problem, for which the gnomes choose the “classic” solution (generally available on the web). HOWEVER…
The eldest gnome unselfishly places himself at the back of the queue; but he is astounded to see the evil king randomly distribute three red hats among the promised black and white attire!
Furthermore, soon after the “game” begins, the king begins to fire his pistol sporadically, whether the current gnome has guessed correctly, or not! There are five such intermittent shots (thankfully in the air).
- Should the appearance of red hats alter the gnomes’ approach?
- Should the king’s staccato shooting alter the gnomes’ approach?